The Rule of 100 ln(2)

The Time Value of Money

Let’s say that you have $100 and you decide to stick it into the bank. This bank is going to pay you some interest for the pleasure of holding onto your money for you. Specifically, at the end of each year, your balance will increase by 5%. Here’s a question that might come to mind: How long will it take for your$100 to double?

The fast-thinker may want to say 20 years, for 20 years of 5% gives you 100% right?

The above is not true. You will earn interest upon interest, and so you will actually double your money in less than 20 years.

This is the magic of compound interest. The internet tells me that Einstein is quoted as saying that “compound interest is the eighth wonder of the world” but I call BS on that one. Happy to be proven wrong though!

To go through how it works in more detail, after your $100 spends one quite boring year in the bank, they will reward you with 5% of your savings which is exactly$5. After another year, they will reward you with another 5% of your savings, but because of your earlier interest payment, 5% of your savings is actually now 5% of $105 and this is$5.25.

This 25c might not seem like much, but let’s keep going. The following year you will get 5% of 110.25 which is $5.5125, so an extra 51c relative to the$5 you are earning on your original $100. The interest your interest earns gets bigger. In fact, you will end up doubling your money in 14 years. You can see how putting money away and earning interest on it can change your retirement dramatically. Let’s say you can find a nice safe place to put your money away for a 5% return. Then it will double every 14 years. This means it will increase by a factor of 8 over 42 years. That’s a pretty good return for not touching it. Note: the time value of money is both a fundamental building block of economics and another contributer to wealth inequality (both because people with zero dollars can not earn interest but also the whole money-creates-money thing means rich people like to lock their wealth up in illiquid assets so they can be squillionaires when they are ninety and so trickling down effects are constrained to trusts and wills). Doubling Time I mentioned that dollars invested at 5% compounded annually would take about 14 years to double but I left out the maths. So let’s cover it more generally here. Suppose we have$$A$ and we invest this at a rate of $r$ compounded annually (in the ongoing example we had $r = 0.05$). How many years will it take to double?

After one year, we receive our first lot of interest and so now have:

$A (1+r)$

After two years, we receive another lot of interest and so now have:

$A (1+r)^2$

After $n$ years, we receive our $n$th lot of interest and so now have:

$A (1+r)^n$

We want to know how long it takes to double our money, so we set the above to $2A$ and then proceed to solve for $n$.

$A (1+r)^n = 2A$

Of course, the $A$ factor (note to self: this is not a bad rap name) on both sides cancels and now we have:

$(1+r)^n = 2$

Taking the natural logarithm of both sides gives us this:

$n \ln(1+r) = \ln 2$

We can now divide through by $\ln(1+r)$ to get

$n = \frac{\ln 2}{\ln(1+r)}.$

This is a solution that a mathematician is happy with but there’s a bit more massaging we can do to get a recipe that one can use on the spot.

For small values of $r$ (and for most examples in the modern world of interest, $r$ will certainly be small!) we know that:

$\ln(1 + r) \approx r$

By this, we mean that the limit of $\ln(1+r)/r$ will tend to 1 as $r \rightarrow 0$.

Exercise. Prove the above limit using whatever tools you have on you.

As we know that $r$ is teeny and we just want something rough, we can replace $\ln(1+r)$ with $r$ in our equation so that it becomes:

$n = \frac{\ln 2}{r}.$

Additionally, it’s much easier to work with, say, 5% than the number 0.05, so we will write $R = 100r$. Our equation then becomes:

$n = \frac{100 \ln 2}{R}.$

Finally, $\ln 2$ is pretty darn close to 0.69, so we may write:

$n = \frac{69}{R}.$

People in the investing world call this the “rule of 69” or something like that. I guess it’s nicer than the “rule of 100 times the natural log of 2”. I’ve also seen the “rule of 72” kicking around as well but who knows what that’s about.

So, if you are locking up some money and receiving $R$% interest, it will take you roughly $69/R$ years for that money to double.